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Introduction to Trigonometric Functions
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Alright class, today weโre diving into the basics of trigonometric functions! Who can tell me what sine, cosine, and tangent represent in a right-angled triangle?
Sine is the opposite over the hypotenuse, right?
Exactly! So if we denote the angle as ฮธ, we have sin(ฮธ) = opposite/hypotenuse. And what about cosine?
That would be adjacent over hypotenuse!
Correct! Cosine is defined as cos(ฮธ) = adjacent/hypotenuse. How about tangent?
Tangent is opposite over adjacent!
"Great job! So, tan(ฮธ) can be represented as opposite/adjacent. Remember this acronym to help:
Reciprocal Functions
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Now let's discuss the reciprocal functions. Who can tell me what the cosecant function is?
Cosecant is the reciprocal of sine, so it would be hypotenuse over opposite!
Thatโs correct! So, cosec(ฮธ) = hypotenuse/opposite. And how about secant and cotangent?
Secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent.
"Exactly! So we have:
Application of Functions in Triangles
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Now, letโs apply our knowledge. If we have a right triangle where the opposite side is 4 units and the hypotenuse is 5 units, whatโs sin(ฮธ)?
Sin(ฮธ) would be 4/5!
Great! And how about the adjacent side if the hypotenuse remains the same? What formula do we use?
We can use the Pythagorean theorem to find the adjacent side.
Exactly! If the hypotenuse is 5 and the opposite side is 4, using aยฒ + bยฒ = cยฒ, we find the adjacent side to be 3 units.
So tan(ฮธ) is 4/3!
Yes! Always utilize these relationships to solve for unknowns in triangles.
Introduction & Overview
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Definitions of Basic Trigonometric Functions
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โข Definitions:
- sin(ฮธ), cos(ฮธ), tan(ฮธ)
- Reciprocal functions: cosec(ฮธ), sec(ฮธ), cot(ฮธ)
Detailed Explanation
In this chunk, we define the three primary trigonometric functions: sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a triangle to the ratios of its sides.
- Sine (sin): The sine of an angle ฮธ in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): The cosine of angle ฮธ is the ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): The tangent of angle ฮธ is the ratio of the opposite side over the adjacent side.
Additionally, we have reciprocal functions;
- Cosecant (cosec): This is the reciprocal of sine, defined as 1/sin(ฮธ).
- Secant (sec): This is the reciprocal of cosine, defined as 1/cos(ฮธ).
- Cotangent (cot): This is the reciprocal of tangent, defined as 1/tan(ฮธ).
Examples & Analogies
Think of a right triangle as a ladder leaning against a wall. The angle at the ground is ฮธ, the hypotenuse is the length of the ladder, the height it reaches on the wall is the opposite side, and the distance from the wall to the base of the ladder is the adjacent side. The sine function helps you understand how high the ladder reaches related to how far away it is from the wall, which helps in real-life scenarios like construction.
Definitions & Key Concepts
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Key Concepts
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Sine: The ratio of the opposite side to the hypotenuse in a right triangle.
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Cosine: The ratio of the adjacent side to the hypotenuse in a right triangle.
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Tangent: The ratio of the opposite side to the adjacent side in a right triangle.
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Reciprocal Functions: Functions that are the inverses of sine, cosine, and tangent.
Examples & Real-Life Applications
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Examples
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In a right triangle with an angle ฮธ, if the opposite side is 4 cm and the hypotenuse is 5 cm, then sin(ฮธ) = 4/5.
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For a right triangle where the adjacent side is 3 cm and the hypotenuse is 5 cm, cos(ฮธ) = 3/5.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
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In a triangle right and true, sine's opposite, cosine's crew, tangent's ratio, all in view!
๐ Fascinating Stories
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Imagine a right triangle with a brave knight. To climb the tallest mountain, he uses sine to find his path up the opposite side, cosine to stick close to the base, and tangent to know where heโs stepping.
๐ง Other Memory Gems
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Remember SOH-CAH-TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
๐ฏ Super Acronyms
Use the acronym RCS for Reciprocal Functions
- R: for Reciprocal
- C: for Cosecant/Secant/Cotangent.
Flash Cards
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Glossary of Terms
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Term: Sine (sin)
Definition:
The ratio of the length of the opposite side to the hypotenuse in a right triangle.
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Term: Cosine (cos)
Definition:
The ratio of the length of the adjacent side to the hypotenuse in a right triangle.
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Term: Tangent (tan)
Definition:
The ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
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Term: Cosecant (cosec)
Definition:
The reciprocal of sine; hypotenuse divided by the opposite side.
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Term: Secant (sec)
Definition:
The reciprocal of cosine; hypotenuse divided by the adjacent side.
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Term: Cotangent (cot)
Definition:
The reciprocal of tangent; adjacent divided by opposite.